Executive Summary : | We intend to investigate distance and resistance matrices furthermore in detail. In particular, we would like to develop elegant formulas for finding the inverse/Moore-Penrose/determinant of distance/resistance matrices of connected graphs (directed/undirected). As the distance/resistance distance has many physical interpretations (like in chemistry, data communication, psychology, modeling of traffic and many more), these formulas will be very useful in understanding the physical phenomenon precisely. Furthermore, these formulas will be very accurate and presented in a closed form that depends only on the number of vertices/degree sequence of the graph. Typical results of this type can be seen in the book titled Graphs and matrices by R. B. Bapat. Another aspect on which we would like to focus on is the spectral properties of distance matrices. Getting interesting results in this direction will enable us to broaden our perspective on some larger class of non-negative matrices, as distance matrices come under a wider class of matrices called Euclidean distance matrices. These matrices have applications in interpolation theory, optimization theory, probability theory (constructing infinitely divisible distributions via infinitely divisible matrices), etc. In our study, we will be emphasizing on obtaining combinatorial interpretation of spectral properties of the distance matrix. For example, given certain graphs, can we say that the number of negative eigenvalues is the number of pendant edges of the graph. These results will be beneficial for further development in combinatorial matrix theory. |